Grup titik

grup geometris simetri (isometri) yang menjaga setidaknya satu titik tetap
Flag of Hong Kong.svg
Bauhinia blakeana bunga di bendera wilayah Hong Kong memiliki simetri C 5 ; bintang di setiap kelopak memiliki simetri D 5 .
Yin and Yang symbol.svg
Yin dan Yang simbol memiliki geometri simetri C 2 dengan warna terbalik

Dalam geometri, grup titik adalah grup geometris simetri (isometri) yang menjaga setidaknya satu titik tetap. Kelompok titik dapat ada dalam ruang Euklides dengan dimensi apa pun, dan setiap kelompok titik dalam dimensi d adalah subkelompok dari grup ortogonal O(d). Kelompok titik dapat direalisasikan sebagai himpunan matriks ortogonal M yang mengubah titik x menjadi titik y :

y = Mx

dimana asal adalah titik tetap. Elemen kelompok titik dapat berupa rotasi (determinan dari M = 1) atau yang lain refleksi, atau rotasi tidak tepat (determinan dari M = −1).

Kelompok titik diskrit di lebih dari satu dimensi datang dalam keluarga tak berhingga, tetapi dari teorema pembatasan kristalografi dan salah satu teorema Bieberbach. setiap jumlah dimensi hanya memiliki jumlah terbatas dari kelompok titik yang simetris di beberapa kisi atau kisi dengan nomor itu. Ini adalah grup titik kristalografi.

Grup titik kiral dan akiral, grup refleksiSunting

Kelompok titik dapat diklasifikasikan ke dalam kelompok kiral (atau rotasi murni) dan kelompok akiral .[1] Gugus kiral adalah subgrup dari grup ortogonal khusus SO( d ): grup ini hanya berisi transformasi ortogonal yang mempertahankan orientasi, yaitu, determinan +1. Gugus akiral juga mengandung transformasi determinan −1. Dalam gugus akiral, transformasi yang mempertahankan orientasi membentuk subgrup (kiral) dari indeks 2.

Grup Coxeter Hingga atau kelompok refleksi adalah kelompok titik yang dihasilkan murni oleh sekumpulan cermin pantul yang melewati titik yang sama. Grup peringkat n Coxeter memiliki mirror n dan diwakili oleh diagram Coxeter-Dynkin. Notasi Coxeter menawarkan notasi tanda kurung yang setara dengan diagram Coxeter, dengan simbol markup untuk rotasi dan grup titik subsimetri lainnya. Kelompok refleksi harus akiral (kecuali untuk grup trivial yang hanya mengandung elemen identitas).

Daftar grup titikSunting

Satu dimensiSunting

Hanya ada dua grup titik satu dimensi, yaitu grup identitas dan kelompok refleksi.

Group Coxeter Diagram Coxeter Urutan Deskripsi
C1 [ ]+ 1 Identity
D1 [ ] CDel node.png 2 Reflection group

Dua dimensiSunting

Grup titik dalam dua dimensi, terkadang disebut grup roset.

Mereka datang dalam dua keluarga yang tidak terbatas:

  1. Grup siklik Cn of n - grup rotasi lipat
  2. Grup dihedral Dn of n-kelompok rotasi dan refleksi lipat

Menerapkan teorema pembatasan kristalografi membatasi n ke nilai 1, 2, 3, 4, dan 6 untuk kedua famili, menghasilkan 10 kelompok.

Grup Intl Orbifold Coxeter Urutan Deskripsi
Cn n n• [n]+ n Siklik: n - rotasi lipat. Grup abstrak Z n , grup bilangan bulat di bawah penambahan modulo n .
Dn nm *n• [n] 2n Dihedral: siklik dengan refleksi. Grup abstrak Dihn, grup dihedral.
Isomorfisme hingga dan korespondensi

Himpunan bagian dari grup titik pantulan murni, yang ditentukan oleh 1 atau 2 mirror, juga dapat diberikan oleh grup Coxeter dan poligon terkait. Ini termasuk 5 kelompok kristalografi. Simetri kelompok pantulan dapat digandakan dengan isomorphism, memetakan kedua cermin satu sama lain dengan cermin membagi dua, menggandakan simetri.

Reflektif Rotasi Terkait
poligon
Group Grup Coxeter Diagram Coxeter Urutan Subgrup Coxeter Urutan
D1 A1 [ ] CDel node.png CDel node c1.png 2 C1 []+ 1 Digon
D2 A12 [2] CDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c2.png 4 C2 [2]+ 2 Persegi panjang
D3 A2 [3] CDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.png 6 C3 [3]+ 3 Segitiga sama sisi
D4 BC2 [4] CDel node.pngCDel 4.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.png 8 C4 [4]+ 4 Persegi
D5 H2 [5] CDel node.pngCDel 5.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c2.png 10 C5 [5]+ 5 Segi lima biasa
D6 G2 [6] CDel node.pngCDel 6.pngCDel node.png CDel node c1.pngCDel 6.pngCDel node c2.png 12 C6 [6]+ 6 Segi enam biasa
Dn I2(n) [n] CDel node.pngCDel n.pngCDel node.png CDel node c1.pngCDel n.pngCDel node c2.png 2n Cn [n]+ n Poligon beraturan
D2×2 A12×2 [[2]] = [4] CDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c1.png = CDel node c1.pngCDel 4.pngCDel node.png 8
D3×2 A2×2 [[3]] = [6] CDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c1.png = CDel node c1.pngCDel 6.pngCDel node.png 12
D4×2 BC2×2 [[4]] = [8] CDel node.pngCDel 4.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c1.png = CDel node c1.pngCDel 8.pngCDel node.png 16
D5×2 H2×2 [[5]] = [10] CDel node.pngCDel 5.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c1.png = CDel node c1.pngCDel 10.pngCDel node.png 20
D6×2 G2×2 [[6]] = [12] CDel node.pngCDel 6.pngCDel node.png CDel node c1.pngCDel 6.pngCDel node c1.png = CDel node c1.pngCDel 12.pngCDel node.png 24
Dn×2 I2(n)×2 [[n]] = [2n] CDel node.pngCDel n.pngCDel node.png CDel node c1.pngCDel n.pngCDel node c1.png = CDel node c1.pngCDel 2x.pngCDel n.pngCDel node.png 4n

Tiga dimensiSunting

Grup titik dalam tiga dimensi, kadang disebut grup titik molekul setelah digunakan secara luas dalam mempelajari kesimetrian molekul kecil.

Mereka datang dalam 7 kelompok tak terbatas dari kelompok aksial atau prismatik, dan 7 kelompok polihedral atau Platonis tambahan. Dalam Notasi Schönflies, *

  • Grup aksial: Cn, S2n, Cnh, Cnv, Dn, Dnd, Dnh
  • Grup polihedral: T, Td, Th, O, Oh, I, Ih

Menerapkan teorema pembatasan kristalografi ke grup ini menghasilkan 32 grup titik kristalografi.

Domain fundamental berwarna genap/ganjil dari grup reflektif
C1v
Urutan 2
C2v
Urutan 4
C3v
Urutan 6
C4v
Urutan 8
C5v
Urutan 10
C6v
Urutan 12
...
Spherical digonal hosohedron2.png Spherical square hosohedron2.png Spherical hexagonal hosohedron2.png Spherical octagonal hosohedron2.png Spherical decagonal hosohedron2.png Spherical dodecagonal hosohedron2.png
D1h
Urutan 4
D2h
Urutan 8
D3h
Urutan 12
D4h
Urutan 16
D5h
Urutan 20
D6h
Urutan 24
...
Spherical digonal bipyramid2.svg Spherical square bipyramid2.svg Spherical hexagonal bipyramid2.png Spherical octagonal bipyramid2.png Spherical decagonal bipyramid2.png Spherical dodecagonal bipyramid2.png
Td
Urutan 24
Oh
Urutan 48
Ih
Urutan 120
Tetrahedral reflection domains.png Octahedral reflection domains.png Icosahedral reflection domains.png
[[Notasi Hermann–Mauguin|Intl]* Geo
[2]
Orbifold Schönflies Conway Coxeter Urutan
1 1 1 C1 C1 [ ]+ 1
1 22 ×1 Ci = S2 CC2 [2+,2+] 2
2 = m 1 *1 Cs = C1v = C1h ±C1 = CD2 [ ] 2
2
3
4
5
6
n
2
3
4
5
6
n
22
33
44
55
66
nn
C2
C3
C4
C5
C6
Cn
C2
C3
C4
C5
C6
Cn
[2]+
[3]+
[4]+
[5]+
[6]+
[n]+
2
3
4
5
6
n
mm2
3m
4mm
5m
6mm
nmm
nm
2
3
4
5
6
n
*22
*33
*44
*55
*66
*nn
C2v
C3v
C4v
C5v
C6v
Cnv
CD4
CD6
CD8
CD10
CD12
CD2n
[2]
[3]
[4]
[5]
[6]
[n]
4
6
8
10
12
2n
2/m
6
4/m
10
6/m
n/m
2n
2 2
3 2
4 2
5 2
6 2
n 2
2*
3*
4*
5*
6*
n*
C2h
C3h
C4h
C5h
C6h
Cnh
±C2
CC6
±C4
CC10
±C6
±Cn / CC2n
[2,2+]
[2,3+]
[2,4+]
[2,5+]
[2,6+]
[2,n+]
4
6
8
10
12
2n
4
3
8
5
12
2n
n
4 2
6 2
8 2
10 2
12 2
2n 2





S4
S6
S8
S10
S12
S2n
CC4
±C3
CC8
±C5
CC12
CC2n / ±Cn
[2+,4+]
[2+,6+]
[2+,8+]
[2+,10+]
[2+,12+]
[2+,2n+]
4
6
8
10
12
2n
Intl Geo Intl Geo Orbifold Schönflies Conway Coxeter Urutan
222
32
422
52
622
n22
n2
2 2
3 2
4 2
5 2
6 2
n 2
222
223
224
225
226
22n
D2
D3
D4
D5
D6
Dn
D4
D6
D8
D10
D12
D2n
[2,2]+
[2,3]+
[2,4]+
[2,5]+
[2,6]+
[2,n]+
4
6
8
10
12
2n
mmm
6m2
4/mmm
10m2
6/mmm
n/mmm
2nm2
2 2
3 2
4 2
5 2
6 2
n 2
*222
*223
*224
*225
*226
*22n
D2h
D3h
D4h
D5h
D6h
Dnh
±D4
DD12
±D8
DD20
±D12
±D2n / DD4n
[2,2]
[2,3]
[2,4]
[2,5]
[2,6]
[2,n]
8
12
16
20
24
4n
42m
3m
82m
5m
122m
2n2m
nm
4 2
6 2
8 2
10 2
12 2
n 2
2*2
2*3
2*4
2*5
2*6
2*n
D2d
D3d
D4d
D5d
D6d
Dnd
±D4
±D6
DD16
±D10
DD24
DD4n / ±D2n
[2+,4]
[2+,6]
[2+,8]
[2+,10]
[2+,12]
[2+,2n]
8
12
16
20
24
4n
23 3 3 332 T T [3,3]+ 12
m3 4 3 3*2 Th ±T [3+,4] 24
43m 3 3 *332 Td TO [3,3] 24
432 4 3 432 O O [3,4]+ 24
m3m 4 3 *432 Oh ±O [3,4] 48
532 5 3 532 I I [3,5]+ 60
53m 5 3 *532 Ih ±I [3,5] 120
(*) Ketika entri Intl digandakan, yang pertama untuk genap n , yang kedua untuk ganjil n .

Grup refleksiSunting

Isomorfisme hingga dan korespondensi

Kelompok titik refleksi, ditentukan oleh 1 sampai 3 bidang cermin, juga dapat diberikan oleh grup Coxeter dan polihedra terkait. Grup [3,3] dapat digandakan, ditulis sebagai [[ 3,3]], memetakan cermin pertama dan terakhir satu sama lain, menggandakan simetri menjadi 48, dan isomorfik ke grup [4,3].

Schönflies grup Coxeter Diagram Coxeter Urutan Terkait reguler dan
Td A3 [3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png 24 Tetrahedron
Td×Dih1 = Oh A3×2 = BC3 [[3,3]] = [4,3] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.png = CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.png 48 Oktahedron bintang
Oh BC3 [4,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.png 48 kubus, segi delapan
Ih H3 [5,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.png 120 Icosahedron, dodecahedron
D3h A2×A1 [3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.png 12 Prisma segitiga
D3h×Dih1 = D6h A2×A1×2 [[3],2] CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c2.png = CDel node c1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node c2.png 24 Prisma segi enam
D4h BC2×A1 [4,2] CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.png 16 Prisma persegi
D4h×Dih1 = D8h BC2×A1×2 [[4],2] = [8,2] CDel node c1.pngCDel 4.pngCDel node c1.pngCDel 2.pngCDel node c2.png = CDel node c1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node c2.png 32 Octagonal prism
D5h H2×A1 [5,2] CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 2.pngCDel node c3.png 20 Prisma segi lima
D6h G2×A1 [6,2] CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 2.pngCDel node c3.png 24 Prisma segi enam
Dnh I2(n)×A1 [n,2] CDel node.pngCDel n.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel n.pngCDel node c2.pngCDel 2.pngCDel node c3.png 4n n-gonal prisma
Dnh×Dih1 = D2nh I2(n)×A1×2 [[n],2] CDel node c1.pngCDel n.pngCDel node c1.pngCDel 2.pngCDel node c2.png = CDel node c1.pngCDel 2x.pngCDel n.pngCDel node.pngCDel 2.pngCDel node c2.png 8n
D2h A13 [2,2] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png 8 Balok
D2h×Dih1 A13×2 [[2],2] = [4,2] CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c2.png 16
D2h×Dih3 = Oh A13×6 [3[2,2]] = [4,3] CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 48
C3v A2 [1,3] CDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.png 6 Hosohedron
C4v BC2 [1,4] CDel node.pngCDel 4.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.png 8
C5v H2 [1,5] CDel node.pngCDel 5.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c2.png 10
C6v G2 [1,6] CDel node.pngCDel 6.pngCDel node.png CDel node c1.pngCDel 6.pngCDel node c2.png 12
Cnv I2(n) [1,n] CDel node.pngCDel n.pngCDel node.png CDel node c1.pngCDel n.pngCDel node c2.png 2n
Cnv×Dih1 = C2nv I2(n)×2 [1,[n]] = [1,2n] CDel node c1.pngCDel n.pngCDel node c1.png = CDel node c1.pngCDel 2x.pngCDel n.pngCDel node.png 4n
C2v A12 [1,2] CDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c2.png 4
C2v×Dih1 A12×2 [1,[2]] CDel node c1.pngCDel 2.pngCDel node c1.png = CDel node c1.pngCDel 4.pngCDel node.png 8
Cs A1 [1,1] CDel node.png CDel node c1.png 2

Four dimensionsSunting

The four-dimensional point groups (chiral as well as achiral) are listed in Conway and Smith,[1] Section 4, Tables 4.1-4.3.

Finite isomorphism and correspondences

The following list gives the four-dimensional reflection groups (excluding those that leave a subspace fixed and that are therefore lower-dimensional reflection groups). Each group is specified as a Coxeter group, and like the polyhedral groups of 3D, it can be named by its related convex regular 4-polytope. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3]+ has three 3-fold gyration points and symmetry order 60. Front-back symmetric groups like [3,3,3] and [3,4,3] can be doubled, shown as double brackets in Coxeter's notation, for example [[3,3,3]] with its order doubled to 240.

Coxeter group/notation Coxeter diagram Order Related polytopes
A4 [3,3,3] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png 120 5-cell
A4×2 [[3,3,3]] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.png 240 5-cell dual compound
BC4 [4,3,3] CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png 384 16-cell/Tesseract
D4 [31,1,1] CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.png 192 Demitesseractic
D4×2 = BC4 <[3,31,1]> = [4,3,3] CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c3.png = CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png 384
D4×6 = F4 [3[31,1,1]] = [3,4,3] CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c1.png = CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 1152
F4 [3,4,3] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c3.pngCDel 3.pngCDel node c4.png 1152 24-cell
F4×2 [[3,4,3]] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c1.png 2304 24-cell dual compound
H4 [5,3,3] CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png 14400 120-cell/600-cell
A3×A1 [3,3,2] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.png 48 Tetrahedral prism
A3×A1×2 [[3,3],2] = [4,3,2] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.png = CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.png 96 Octahedral prism
BC3×A1 [4,3,2] CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.png 96
H3×A1 [5,3,2] CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.png 240 Icosahedral prism
A2×A2 [3,2,3] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 3.pngCDel node c4.png 36 Duoprism
A2×BC2 [3,2,4] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 4.pngCDel node c4.png 48
A2×H2 [3,2,5] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 5.pngCDel node c4.png 60
A2×G2 [3,2,6] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png 72
BC2×BC2 [4,2,4] CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 4.pngCDel node c4.png 64
BC22×2 [[4,2,4]] CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node c1.png 128
BC2×H2 [4,2,5] CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 5.pngCDel node c4.png 80
BC2×G2 [4,2,6] CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png 96
H2×H2 [5,2,5] CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 5.pngCDel node c4.png 100
H2×G2 [5,2,6] CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png 120
G2×G2 [6,2,6] CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png 144
I2(p)×I2(q) [p,2,q] CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel q.pngCDel node c4.png 4pq
I2(2p)×I2(q) [[p],2,q] = [2p,2,q] CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel q.pngCDel node c3.png = CDel node c1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel q.pngCDel node c3.png 8pq
I2(2p)×I2(2q) [[p]],2,[[q]] = [2p,2,2q] CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel q.pngCDel node c2.png = CDel node c1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 2x.pngCDel q.pngCDel node.png 16pq
I2(p)2×2 [[p,2,p]] CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c2.pngCDel p.pngCDel node c1.png 8p2
I2(2p)2×2 [[[p],2,[p]]] = [[2p,2,2p]] CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel p.pngCDel node c1.png = CDel node c1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node c1.pngCDel 2x.pngCDel p.pngCDel node.png 32p2
A2×A1×A1 [3,2,2] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png 24
BC2×A1×A1 [4,2,2] CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png 32
H2×A1×A1 [5,2,2] CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png 40
G2×A1×A1 [6,2,2] CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png 48
I2(p)×A1×A1 [p,2,2] CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png 8p
I2(2p)×A1×A1×2 [[p],2,2] = [2p,2,2] CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png = CDel node c1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png 16p
I2(p)×A12×2 [p,2,[2]] = [p,2,4] CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c3.png = CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 4.pngCDel node.png 16p
I2(2p)×A12×4 [[p]],2,[[2]] = [2p,2,4] CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c2.png = CDel node c1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node.png 32p
A1×A1×A1×A1 [2,2,2] CDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png 16 4-orthotope
A12×A1×A1×2 [[2],2,2] = [4,2,2] CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png 32
A12×A12×4 [[2]],2,[[2]] = [4,2,4] CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c2.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node.png 64
A13×A1×6 [3[2,2],2] = [4,3,2] CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node c2.png 96
A14×24 [3,3[2,2,2]] = [4,3,3] CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 384

Five dimensionsSunting

Finite isomorphism and correspondences

The following table gives the five-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3]+ has four 3-fold gyration points and symmetry order 360.

Coxeter group/notation Coxeter
diagrams
Order Related regular and
prismatic polytopes
A5 [3,3,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.png 720 5-simplex
A5×2 [[3,3,3,3]] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.png 1440 5-simplex dual compound
BC5 [4,3,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.png 3840 5-cube, 5-orthoplex
D5 [32,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.png 1920 5-demicube
D5×2 <[3,3,31,1]> CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png = CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png 3840
A4×A1 [3,3,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 2.pngCDel node c5.png 240 5-cell prism
A4×A1×2 [[3,3,3],2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.png 480
BC4×A1 [4,3,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 2.pngCDel node c5.png 768 tesseract prism
F4×A1 [3,4,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 2.pngCDel node c5.png 2304 24-cell prism
F4×A1×2 [[3,4,3],2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.png 4608
H4×A1 [5,3,3,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 2.pngCDel node c5.png 28800 600-cell or 120-cell prism
D4×A1 [31,1,1,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 2.pngCDel node c5.png 384 Demitesseract prism
A3×A2 [3,3,2,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 3.pngCDel node c5.png 144 Duoprism
A3×A2×2 [[3,3],2,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel 3.pngCDel node c4.png 288
A3×BC2 [3,3,2,4] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 4.pngCDel node c5.png 192
A3×H2 [3,3,2,5] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 5.pngCDel node c5.png 240
A3×G2 [3,3,2,6] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 6.pngCDel node c5.png 288
A3×I2(p) [3,3,2,p] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel p.pngCDel node c5.png 48p
BC3×A2 [4,3,2,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 3.pngCDel node c5.png 288
BC3×BC2 [4,3,2,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 4.pngCDel node c5.png 384
BC3×H2 [4,3,2,5] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 5.pngCDel node c5.png 480
BC3×G2 [4,3,2,6] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 6.pngCDel node c5.png 576
BC3×I2(p) [4,3,2,p] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel p.pngCDel node c5.png 96p
H3×A2 [5,3,2,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 3.pngCDel node c5.png 720
H3×BC2 [5,3,2,4] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 4.pngCDel node c5.png 960
H3×H2 [5,3,2,5] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 5.pngCDel node c5.png 1200
H3×G2 [5,3,2,6] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 6.pngCDel node c5.png 1440
H3×I2(p) [5,3,2,p] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 240p
A3×A12 [3,3,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 96
BC3×A12 [4,3,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 192
H3×A12 [5,3,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 480
A22×A1 [3,2,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 72 duoprism prism
A2×BC2×A1 [3,2,4,2] CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png 96
A2×H2×A1 [3,2,5,2] CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png 120
A2×G2×A1 [3,2,6,2] CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png 144
BC22×A1 [4,2,4,2] CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png 128
BC2×H2×A1 [4,2,5,2] CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png 160
BC2×G2×A1 [4,2,6,2] CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png 192
H22×A1 [5,2,5,2] CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png 200
H2×G2×A1 [5,2,6,2] CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png 240
G22×A1 [6,2,6,2] CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png 288
I2(p)×I2(q)×A1 [p,2,q,2] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.png 8pq
A2×A13 [3,2,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 48
BC2×A13 [4,2,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 64
H2×A13 [5,2,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 80
G2×A13 [6,2,2,2] CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 96
I2(p)×A13 [p,2,2,2] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 16p
A15 [2,2,2,2] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 2.pngCDel node c5.png 32 5-orthotope
A15×(2!) [[2],2,2,2] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png 64
A15×(2!×2!) [[2]],2,[2],2] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c3.png 128
A15×(3!) [3[2,2],2,2] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png 192
A15×(3!×2!) [3[2,2],2,[[2]] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c2.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node.png 384
A15×(4!) [3,3[2,2,2],2]] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node c2.png 768
A15×(5!) [3,3,3[2,2,2,2]] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 3840

Six dimensionsSunting

Finite isomorphism and correspondences

The following table gives the six-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3]+ has five 3-fold gyration points and symmetry order 2520.

Coxeter group Coxeter
diagram
Order Related regular and
prismatic polytopes
A6 [3,3,3,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 5040 (7!) 6-simplex
A6×2 [[3,3,3,3,3]] CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png 10080 (2×7!) 6-simplex dual compound
BC6 [4,3,3,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 46080 (26×6!) 6-cube, 6-orthoplex
D6 [3,3,3,31,1] CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 23040 (25×6!) 6-demicube
E6 [3,32,2] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 51840 (72×6!) 122, 221
A5×A1 [3,3,3,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 1440 (2×6!) 5-simplex prism
BC5×A1 [4,3,3,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 7680 (26×5!) 5-cube prism
D5×A1 [3,3,31,1,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 3840 (25×5!) 5-demicube prism
A4×I2(p) [3,3,3,2,p] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 240p Duoprism
BC4×I2(p) [4,3,3,2,p] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 768p
F4×I2(p) [3,4,3,2,p] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 2304p
H4×I2(p) [5,3,3,2,p] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 28800p
D4×I2(p) [3,31,1,2,p] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 384p
A4×A12 [3,3,3,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 480
BC4×A12 [4,3,3,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 1536
F4×A12 [3,4,3,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 4608
H4×A12 [5,3,3,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 57600
D4×A12 [3,31,1,2,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 768
A32 [3,3,2,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 576
A3×BC3 [3,3,2,4,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 1152
A3×H3 [3,3,2,5,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png 2880
BC32 [4,3,2,4,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 2304
BC3×H3 [4,3,2,5,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png 5760
H32 [5,3,2,5,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png 14400
A3×I2(p)×A1 [3,3,2,p,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png 96p Duoprism prism
BC3×I2(p)×A1 [4,3,2,p,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png 192p
H3×I2(p)×A1 [5,3,2,p,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png 480p
A3×A13 [3,3,2,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 192
BC3×A13 [4,3,2,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 384
H3×A13 [5,3,2,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 960
I2(p)×I2(q)×I2(r) [p,2,q,2,r] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel r.pngCDel node.png 8pqr Triaprism
I2(p)×I2(q)×A12 [p,2,q,2,2] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 16pq
I2(p)×A14 [p,2,2,2,2] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 32p
A16 [2,2,2,2,2] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 64 6-orthotope

Seven dimensionsSunting

The following table gives the seven-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3,3]+ has six 3-fold gyration points and symmetry order 20160.

Coxeter group Coxeter diagram Order Related polytopes
A7 [3,3,3,3,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 40320 (8!) 7-simplex
A7×2 [[3,3,3,3,3,3]] CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png 80640 (2×8!) 7-simplex dual compound
BC7 [4,3,3,3,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 645120 (27×7!) 7-cube, 7-orthoplex
D7 [3,3,3,3,31,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 322560 (26×7!) 7-demicube
E7 [3,3,3,32,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 2903040 (8×9!) 321, 231, 132
A6×A1 [3,3,3,3,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 10080 (2×7!)
BC6×A1 [4,3,3,3,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 92160 (27×6!)
D6×A1 [3,3,3,31,1,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 46080 (26×6!)
E6×A1 [3,3,32,1,2] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea.png 103680 (144×6!)
A5×I2(p) [3,3,3,3,2,p] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 1440p
BC5×I2(p) [4,3,3,3,2,p] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 7680p
D5×I2(p) [3,3,31,1,2,p] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 3840p
A5×A12 [3,3,3,3,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 2880
BC5×A12 [4,3,3,3,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 15360
D5×A12 [3,3,31,1,2,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 7680
A4×A3 [3,3,3,2,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 2880
A4×BC3 [3,3,3,2,4,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 5760
A4×H3 [3,3,3,2,5,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png 14400
BC4×A3 [4,3,3,2,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 9216
BC4×BC3 [4,3,3,2,4,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 18432
BC4×H3 [4,3,3,2,5,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png 46080
H4×A3 [5,3,3,2,3,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 345600
H4×BC3 [5,3,3,2,4,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 691200
H4×H3 [5,3,3,2,5,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png 1728000
F4×A3 [3,4,3,2,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 27648
F4×BC3 [3,4,3,2,4,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 55296
F4×H3 [3,4,3,2,5,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png 138240
D4×A3 [31,1,1,2,3,3] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 4608
D4×BC3 [3,31,1,2,4,3] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 9216
D4×H3 [3,31,1,2,5,3] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png 23040
A4×I2(p)×A1 [3,3,3,2,p,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png 480p
BC4×I2(p)×A1 [4,3,3,2,p,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png 1536p
D4×I2(p)×A1 [3,31,1,2,p,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png 768p
F4×I2(p)×A1 [3,4,3,2,p,2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png 4608p
H4×I2(p)×A1 [5,3,3,2,p,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png 57600p
A4×A13 [3,3,3,2,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 960
BC4×A13 [4,3,3,2,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 3072
F4×A13 [3,4,3,2,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 9216
H4×A13 [5,3,3,2,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 115200
D4×A13 [3,31,1,2,2,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 1536
A32×A1 [3,3,2,3,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 1152
A3×BC3×A1 [3,3,2,4,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 2304
A3×H3×A1 [3,3,2,5,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 5760
BC32×A1 [4,3,2,4,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 4608
BC3×H3×A1 [4,3,2,5,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 11520
H32×A1 [5,3,2,5,3,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 28800
A3×I2(p)×I2(q) [3,3,2,p,2,q] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png 96pq
BC3×I2(p)×I2(q) [4,3,2,p,2,q] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png 192pq
H3×I2(p)×I2(q) [5,3,2,p,2,q] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png 480pq
A3×I2(p)×A12 [3,3,2,p,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 192p
BC3×I2(p)×A12 [4,3,2,p,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 384p
H3×I2(p)×A12 [5,3,2,p,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 960p
A3×A14 [3,3,2,2,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 384
BC3×A14 [4,3,2,2,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 768
H3×A14 [5,3,2,2,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 1920
I2(p)×I2(q)×I2(r)×A1 [p,2,q,2,r,2] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel r.pngCDel node.pngCDel 2.pngCDel node.png 16pqr
I2(p)×I2(q)×A13 [p,2,q,2,2,2] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 32pq
I2(p)×A15 [p,2,2,2,2,2] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 64p
A17 [2,2,2,2,2,2] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 128

Eight dimensionsSunting

The following table gives the eight-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3,3,3]+ has seven 3-fold gyration points and symmetry order 181440.

Coxeter group Coxeter diagram Order Related polytopes
A8 [3,3,3,3,3,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 362880 (9!) 8-simplex
A8×2 [[3,3,3,3,3,3,3]] CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png 725760 (2×9!) 8-simplex dual compound
BC8 [4,3,3,3,3,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 10321920 (288!) 8-cube,8-orthoplex
D8 [3,3,3,3,3,31,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 5160960 (278!) 8-demicube
E8 [3,3,3,3,32,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 696729600 (192×10!) 421, 241, 142
A7×A1 [3,3,3,3,3,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 80640 7-simplex prism
BC7×A1 [4,3,3,3,3,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 645120 7-cube prism
D7×A1 [3,3,3,3,31,1,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png 322560 7-demicube prism
E7×A1 [3,3,3,32,1,2] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea.png 5806080 321 prism, 231 prism, 142 prism
A6×I2(p) [3,3,3,3,3,2,p] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 10080p duoprism
BC6×I2(p) [4,3,3,3,3,2,p] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 92160p
D6×I2(p) [3,3,3,31,1,2,p] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 46080p
E6×I2(p) [3,3,32,1,2,p] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png 103680p
A6×A12 [3,3,3,3,3,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 20160
BC6×A12 [4,3,3,3,3,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 184320
D6×A12 [33,1,1,2,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 92160
E6×A12 [3,3,32,1,2,2] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea.pngCDel 2.pngCDel nodea.png 207360
A5×A3 [3,3,3,3,2,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 17280
BC5×A3 [4,3,3,3,2,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 92160
D5×A3 [32,1,1,2,3,3] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 46080
A5×BC3 [3,3,3,3,2,4,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 34560
BC5×BC3 [4,3,3,3,2,4,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 184320
D5×BC3 [32,1,1,2,4,3] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 92160
A5×H3 [3,3,3,3,2,5,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
BC5×H3 [4,3,3,3,2,5,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
D5×H3 [32,1,1,2,5,3] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
A5×I2(p)×A1 [3,3,3,3,2,p,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png
BC5×I2(p)×A1 [4,3,3,3,2,p,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png
D5×I2(p)×A1 [32,1,1,2,p,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png
A5×A13 [3,3,3,3,2,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
BC5×A13 [4,3,3,3,2,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
D5×A13 [32,1,1,2,2,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
A4×A4 [3,3,3,2,3,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
BC4×A4 [4,3,3,2,3,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
D4×A4 [31,1,1,2,3,3,3] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
F4×A4 [3,4,3,2,3,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
H4×A4 [5,3,3,2,3,3,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
BC4×BC4 [4,3,3,2,4,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
D4×BC4 [31,1,1,2,4,3,3] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
F4×BC4 [3,4,3,2,4,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
H4×BC4 [5,3,3,2,4,3,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
D4×D4 [31,1,1,2,31,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
F4×D4 [3,4,3,2,31,1,1] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
H4×D4 [5,3,3,2,31,1,1] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
F4×F4 [3,4,3,2,3,4,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
H4×F4 [5,3,3,2,3,4,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
H4×H4 [5,3,3,2,5,3,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
A4×A3×A1 [3,3,3,2,3,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png duoprism prisms
A4×BC3×A1 [3,3,3,2,4,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
A4×H3×A1 [3,3,3,2,5,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
BC4×A3×A1 [4,3,3,2,3,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
BC4×BC3×A1 [4,3,3,2,4,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
BC4×H3×A1 [4,3,3,2,5,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
H4×A3×A1 [5,3,3,2,3,3,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
H4×BC3×A1 [5,3,3,2,4,3,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
H4×H3×A1 [5,3,3,2,5,3,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
F4×A3×A1 [3,4,3,2,3,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
F4×BC3×A1 [3,4,3,2,4,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
F4×H3×A1 [3,4,2,3,5,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
D4×A3×A1 [31,1,1,2,3,3,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
D4×BC3×A1 [31,1,1,2,4,3,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
D4×H3×A1 [31,1,1,2,5,3,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
A4×I2(p)×I2(q) [3,3,3,2,p,2,q] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png triaprism
BC4×I2(p)×I2(q) [4,3,3,2,p,2,q] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png
F4×I2(p)×I2(q) [3,4,3,2,p,2,q] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png
H4×I2(p)×I2(q) [5,3,3,2,p,2,q] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png
D4×I2(p)×I2(q) [31,1,1,2,p,2,q] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png
A4×I2(p)×A12 [3,3,3,2,p,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
BC4×I2(p)×A12 [4,3,3,2,p,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
F4×I2(p)×A12 [3,4,3,2,p,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
H4×I2(p)×A12 [5,3,3,2,p,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
D4×I2(p)×A12 [31,1,1,2,p,2,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
A4×A14 [3,3,3,2,2,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
BC4×A14 [4,3,3,2,2,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
F4×A14 [3,4,3,2,2,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
H4×A14 [5,3,3,2,2,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
D4×A14 [31,1,1,2,2,2,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
A3×A3×I2(p) [3,3,2,3,3,2,p] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png
BC3×A3×I2(p) [4,3,2,3,3,2,p] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png
H3×A3×I2(p) [5,3,2,3,3,2,p] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png
BC3×BC3×I2(p) [4,3,2,4,3,2,p] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png
H3×BC3×I2(p) [5,3,2,4,3,2,p] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png
H3×H3×I2(p) [5,3,2,5,3,2,p] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png
A3×A3×A12 [3,3,2,3,3,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
BC3×A3×A12 [4,3,2,3,3,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
H3×A3×A12 [5,3,2,3,3,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
BC3×BC3×A12 [4,3,2,4,3,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
H3×BC3×A12 [5,3,2,4,3,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
H3×H3×A12 [5,3,2,5,3,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
A3×I2(p)×I2(q)×A1 [3,3,2,p,2,q,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.png
BC3×I2(p)×I2(q)×A1 [4,3,2,p,2,q,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.png
H3×I2(p)×I2(q)×A1 [5,3,2,p,2,q,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.png
A3×I2(p)×A13 [3,3,2,p,2,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
BC3×I2(p)×A13 [4,3,2,p,2,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
H3×I2(p)×A13 [5,3,2,p,2,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
A3×A15 [3,3,2,2,2,2,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
BC3×A15 [4,3,2,2,2,2,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
H3×A15 [5,3,2,2,2,2,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
I2(p)×I2(q)×I2(r)×I2(s) [p,2,q,2,r,2,s] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel r.pngCDel node.pngCDel 2.pngCDel node.pngCDel s.pngCDel node.png 16pqrs
I2(p)×I2(q)×I2(r)×A12 [p,2,q,2,r,2,2] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel r.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 32pqr
I2(p)×I2(q)×A14 [p,2,q,2,2,2,2] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 64pq
I2(p)×A16 [p,2,2,2,2,2,2] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 128p
A18 [2,2,2,2,2,2,2] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png 256

Lihat pulaSunting

ReferensiSunting

  1. ^ a b Conway, John H.; Smith, Derek A. (2003). On quaternions and octonions: their geometry, arithmetic, and symmetry. A K Peters. ISBN 978-1-56881-134-5. 
  2. ^ Grup Ruang Kristalografi dalam aljabar geometris, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF [1]
  • H. S. M. Coxeter: Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [2]
    • (Paper 23) H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
  • H. S. M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, Springer-Verlag. New York. 1980
  • N. W. Johnson: Geometries and Transformations, (2018) Chapter 11: Finite symmetry groups

Pranala luarSunting